Question

On Z an operation * is defined by a * b = a² + b² for all a, b ∈ Z. The operation * on Z is
(a) commutative and associative
(b) associative but not commutative
(c) not associative
(d) not a binary operation

Answer 1

(c) not associative

Commutativity:
$$\begin{gathered} \text{Let}a,b\in Z.\mathrm{Then}, \\ a*b=a^2+b^2 \\ =b^2+a^2 \\ =b*a \\ \text{Therefore,} \\ a*bb*a,\forall a,b\in Z \end{gathered}$$

Thus, * is commutative on Z.

Associativity:
$$\begin{gathered} \text{Let}a,b,c\in Z \\ a*\left(b*c\right)=a*\left(b^2+c^2\right) \\ =a^2+{\left(b^2+c^2\right)}^2 \\ =a^2+b^4+c^4+2b^2{c}^2 \\ \left(a*b\right)*c=\left(a^2+b^2\right)*c \\ ={\left(a^2+b^2\right)}^2+c^2 \\ =a^4+b^4+2a^2{b}^2+c^2 \\ \text{Therefore,} \\ a*\left(b*c\right)\ne \left(a*b\right)*c \end{gathered}$$

Thus, * is not associative on Z.